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Document an algorithm by Fukushima #2246
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documentation/Fukushima.tex
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\subsection*{Definitions} | ||
Jacobi's nome $q\of{m}$ is defined as a function of the elliptic integral $K\of{m}$ as: | ||
\[ | ||
q\of{m} = \exp\of{\frac{-\gp K\of{1 - m}}{K\of{m}}} |
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\DefineAs
, for ≔.
Same in other places below.
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Use \Pi
for the constant π (typeset in roman as a standard symbol, in ISO 80000 style). \gp
is 𝜋, e.g., "let 𝜋 be a permutation...".
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Using \DefineAs
in a bunch of places, please check that it makes sense.
Changed to use \Pi
.
documentation/Fukushima.tex
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\] | ||
We have: $K\of{m} = K_X\of{m_c} X\of{m_c}$. | ||
|
||
An expression for $E\of{m}$ can be obtained from Legendre's relation ([Fuku11a] equation 2.11): |
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E
, F
, K
, and Π
are standard notation, and should be typeset in roman. Add them to semtex.sty
in the vicinity of LegendrePolynomial
, e.g.,
newcommand{\EllipticE}{\operatorname{\StandardSymbol E}}
Keep E_X
for the ad hoc terms, as these are not standard notation.
Similarly B
and D
are fine, since they are Fukushima’s rather than a widespread notation.
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Nope, both Fukushima and NIST use italics for E, F, etc. The Jacobi functions, on the other hand, are in roman but there aren't any in this paper.
\] | ||
These formulæ make it possible, by computing a Maclaurin approximation of $B_X^*\of{m}$ and $E_X\of{m}$, to evaluate $B\of{m}$ | ||
and $D\of{m}$ for $m$ close to $0$. | ||
\printbibliography |
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Cite the code as well?
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Done, kind-of, there appears to be an @Software
tag.
&= \frac{1}{m}\pa{X\of{m_c}\pa{K_X\of{m_c} - E_X\of{m_c}} - \frac{1}{2 K_X\of{m_c}}} | ||
\end{align*} | ||
These formulæ provide a means to compute $B\of{m}$ and $D\of{m}$ for $m$ close to $1$. First, a polynomial approximation of $q\of{m_c}$ | ||
is computed, whose leading term is of order $m_c / 16$. Then the $\log$ of that approximation is evaluated, yielding $X\of{m_c}$ (this |
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Does $\frac {m_c} {16}$
look bad?
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Yeah, it's better that way.
B\of{m} = -\frac{\gp}{m} B_X^*\of{m} | ||
\] | ||
These formulæ make it possible, by computing a Maclaurin approximation of $B_X^*\of{m}$ and $E_X\of{m}$, to evaluate $B\of{m}$ | ||
and $D\of{m}$ for $m$ close to $0$. |
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Consider adding a one-sentence conclusion comparing this method with the one described by Fukushima, so that we understand why this is done that way.
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Added a "Conclusion" section.
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As discussed, clarify the conclusion to actually explain the differences between the two methods.
Also clean up some old files that were still around.